
Convergence follows since \(L\)  is slowly varying on infinity: \( {L(N)}/{L(N\tau )}\rightarrow 1\) . By local uniform convergence (proposition 0.5 in [1]}) \( {L(N) }/{L\left( N\tau {(\log (T)+1)}/{\log (T)} \right)} \rightarrow 1\) .
